A052432 - OEIS

%I #35 Oct 01 2020 12:58:06

%S 11,37,389,5077,234446

%N Smallest conductor of elliptic curve with rank n.

%C The smallest known conductors for ranks 5, 6, 7, and 11 are 19047851, 5187563742, 382623908456, 18031737725935636520843, respectively. These are just upper bounds on a(n).

%H J. E. Cremona, <a href="http://johncremona.github.io/ecdata/">Elliptic Curve Data</a>

%H J. E. Cremona, <a href="http://homepages.warwick.ac.uk/staff/J.E.Cremona/conductors.htm">Best known conductors for elliptic curves of given rank</a>

%H Noam D. Elkies and Mark Watkins, <a href="https://arxiv.org/abs/math/0403374">Elliptic curves of large rank and small conductor</a>, arXiv:math/0403374 [math.NT], 2004.

%H M. O. Rubenstein, <a href="https://arxiv.org/abs/1307.0420v3">Elliptic curves of high rank and the Riemann zeta function</a>, arXiv:1307.0420 [math.NT], 2013.

%e From _Michael Somos_, Apr 12 2012: (Start)

%e The curve "11a3": y^2 + y = x^3 - x^2 has rank 0.

%e The curve "37a1": y^2 + y = x^3 - x has rank 1 with generator [0, 0].

%e The curve "389a1": y^2 + y = x^3 + x^2 - 2 * x has rank 2 with generators [0, 0], [-1, 1].

%e The curve "5077a1": y^2 + y = x^3 - 7 * x + 6 has rank 3 with generators [0, 2], [-1, 3], [-2, 3]. (End)

%K nonn,nice,hard,more

%O 0,1

%A Jesper Petersen (u943254(AT)daimi.au.dk), Mar 14 2000

%E Added value for rank 4 from Cremona's extended tables, by _John Cremona_, Apr 02 2012

%E Upper bounds for a(5)-a(7) and a(11) from Elkies & Watkins added by _Jonathan Sondow_, Oct 29 2013.

%E Unproved values for a(5)-a(7) and a(11) removed by _N. J. A. Sloane_, Jan 25 2016 at the suggestion of John Cremona.